The usual answer I hear is because power needs to be conserved, ...

This is correct.

Normally if I were to double a supply voltage in a circuit, then I would expect twice the current (assuming its a linear circuit).

That would be correct if you didn't change the load resistance.

But if i have a 2x transformer ... then the output of the transformer will have twice the voltage and half the current.

This is true for a given output power.

To maintain output power for a doubling of output voltage the load resistance would be increased by a factor of 4 (the step-up ratio squared).

What exactly is going on electromagnetically that allows a transformer to violate ohms law and give lower currents with higher voltage?

Nothing.

The usual answer I hear is because power needs to be conserved, ...

This is correct.

Normally if I were to double a supply voltage in a circuit, then I would expect twice the current (assuming its a linear circuit).

That would be correct if you didn't change the load resistance.

But if i have a 2x transformer ... then the output of the transformer will have twice the voltage and half the current.

This is true for a given output power.

^{simulate this circuit – Schematic created using CircuitLab}

Figure 1. (a) Reference setup. (b) Doubling the voltage quadruples the power into the same resistance. (c) By quadrupling the load resistance the power is the same for double the voltage.

To maintain output power for a doubling of output voltage the load resistance would be increased by a factor of 4 (the step-up ratio squared).

What exactly is going on electromagnetically that allows a transformer to violate ohms law and give lower currents with higher voltage?

Nothing.

The usual answer I hear is because power needs to be conserved, ...

This is correct.

Normally if I were to double a supply voltage in a circuit, then I would expect twice the current (assuming its a linear circuit).

That would be correct if you didn't change the load resistance.

But if i have a 2x transformer ... then the output of the transformer will have twice the voltage and half the current.

This is true for a given output power. To maintain output power for a doubling of output voltage the load resistance would be increased by a factor of \$ \sqrt 2 \$.

What exactly is going on electromagnetically that allows a transformer to violate ohms law and give lower currents with higher voltage?

Nothing.

The usual answer I hear is because power needs to be conserved, ...

This is correct.

Normally if I were to double a supply voltage in a circuit, then I would expect twice the current (assuming its a linear circuit).

That would be correct if you didn't change the load resistance.

But if i have a 2x transformer ... then the output of the transformer will have twice the voltage and half the current.

This is true for a given output power. 

To maintain output power for a doubling of output voltage the load resistance would be increased by a factor of 4 (the step-up ratio squared).

What exactly is going on electromagnetically that allows a transformer to violate ohms law and give lower currents with higher voltage?

Nothing.